WA6. Applications of Derivatives, Extreme Values of Functions, and the Mean Value Theorem¶

Question 1¶

1. An airplane is flying towards a radar station at a constant height of 6 km above the ground. If the distance s between the airplane and the radar station is decreasing at a rate of 400 km per hour when s=10 km., what is the horizontal speed of the plane? Make sure your answer includes units.

Question 2¶

2. A boat is being pulled into a dock by a rope attached to it and passing through a pulley on the deck, positioned 6 meters higher than the boat. If the rope is being pulled in at a rate of 3 meters/sec, how fast is the boat approaching the dock when it is 8 meters from the dock? Make sure your answer includes units.

Question 3¶

3. A girl is flying a kite on a string. The kite is 120 ft. above the ground and the wind is blowing the kite horizontally away from her at 6 ft/sec. At what rate must she let out the string when 130 ft of string has been released? Make sure your answer includes units.

Question 4¶

4. Find the Linearization of f(x) =sin x at a= \frac{ \pi}{2} . Provide your answer as L(x) = ?.

Question 5¶

5. Use Linear Approximation to estimate e^{(-0.01)} . Provide your answer in 2 decimal places. Do not use a calculator. Show work for credit.

Question 6¶

6. Calculate the locations of maximums and minimums of the following functions: Show work in details.

f(x) = x^3 - 3x + 2
f(x) = x^4 - 8x^2 + 3

Question 7¶

7. Find the exact x-value where the function f(x)=x+ln ( x^2-1 ) attains a maximum value. An estimated answer or a calculator answer will not earn any credit.

Question 8¶

8. Using the Mean Value Theorem and Rolle’s Theorem, show that x^3+ x - 1= 0 has exactly one real root.

Question 9¶

9. If f(1)=10 and f’(x) \geq 2 for 1 \leq x \leq 4 , how small can f(4) possibly be?.

Question 10¶

10. The graph of a function is given below. Determine the intervals on which the function is concave up and concave down.